The correct answer is option (C):
100
Let's break down this geometry problem step by step to find the correct answer.
We're given triangle DEF with points A, B, and C on sides DE, DF, and EF respectively. We also know that EC = AC and CF = BC, and angle D is 40 degrees. Our goal is to find the measure of angle ACB.
First, consider triangle ABC and the given information: EC = AC, so triangle AEC is an isosceles triangle. This means angle CAE = angle CEA. Similarly, CF = BC, making triangle BFC isosceles, and thus angle CBF = angle CFB.
Now, let's look at angles within the main triangle DEF. We have angle D = 40 degrees. The sum of the angles in triangle DEF is 180 degrees. Therefore, angle E + angle F = 180 - 40 = 140 degrees.
Let's denote angle CEA (which is also angle CAE) as 'x'. Since angle CEA is an exterior angle of triangle BEC, angle BEC = 180 - x.
Also, angle ECF = angle CFB. Since angle F = angle ECF + angle ECB, and since CFB = 180 - CEF = 180 - x, then angle F must be part of an expression where angle ECB (the angle made by side BC and side CE) = angle F - angle BCF.
In triangle ACB, the sum of the angles must be 180 degrees:
angle A + angle B + angle ACB = 180
Since A is equal to x, and B equals another expression, we can rewrite the equation. However, this is not an efficient way to solve this problem.
Alternatively, consider the angles around point C. Angle ACB + angle ACE + angle BCF = 180 degrees.
We are looking for angle ACB.
Because EC = AC, then angle E = angle CAE.
Because CF = BC, then angle F = angle CBF.
angle A + angle B + angle D = 180 degrees
Let angle E = a and angle F = b.
Therefore, a + b + 40 = 180 ⇒ a + b = 140.
The angle A = 180 - 2a and the angle B = 180 - 2b.
The sum of angles in ACB:
angle ACB + (180 - 2a) + (180 - 2b) = 180
angle ACB = 2a + 2b - 180
angle ACB = 2(a + b) - 180
angle ACB = 2(140) - 180
angle ACB = 280 - 180
angle ACB = 100 degrees
Therefore, the measure of angle ACB is 100 degrees.